Extendibility of Latin Hypercuboids

TL;DR

This paper investigates the conditions under which Latin hypercuboids can be extended or completed to Latin hypercubes, introduces a generalized array framework, and provides constructions demonstrating limitations in extendibility.

Contribution

It characterizes extendibility and completability of Latin hypercuboids, generalizes to multidimensional set arrays, and constructs examples showing non-extendibility.

Findings

Certain Latin hypercuboids cannot be extended to hypercubes.

A generalized array construction does not always allow completability.

Existing constructions like Pebody's cannot be used for the new arrays.

Abstract

A Latin hypercuboid of order $n$ is a $d$-dimensional matrix of dimensions $n\times n\times\cdots\times n\times k$, with symbols from a set of cardinality $n$ such that each symbol occurs at most once in each axis-parallel line. If $k=n$ the hypercuboid is a Latin hypercube. The Latin hypercuboid is \emph{completable} if it is contained in a Latin hypercube of the same order and dimension. It is \emph{extendible} if it can have one extra layer added. In this note we consider which Latin hypercuboids are completable/extendible. We also consider a generalisation that involves multidimensional arrays of sets that satisfy certain balance properties. The extendibility problem corresponds to choosing representatives from the sets in a way that is analogous to a choice of a Hall system of distinct representatives, but in higher dimensions. The completability problem corresponds to partitioning the sets into such SDRs. We provide a construction for such an array of sets that does not have the property analogous to completability. A related concept was introduced by H\"aggkvist under the name $(m,m,m)$-array. We generalise a construction of $(m,m,m)$-arrays credited to Pebody, but show that it cannot be used to build the arrays that we need.